On my way home today I saw an ad for condos and it said 50% sold out. Does this mean that there's a 50% chance you'll have a neighbour if you bought one and moved in, or is it more?

Zeno, feel free to move to probability, if more appropriate. I know how much you love exercising mod powers and move threads. lol

If exactly 50% of the condos have been sold, there are n sold condos and n unsold condos. Are the condos arranged in a line, such that the condos on each end are neighboring one other condo, and all the rest of the condos in the middle neighbor two condos? Should we assume the sold condos follow a uniform random distribution throughout the complex? Seems doubtful: there are probably areas of the complex that are more desirable than others (ie purchasers prefer being farther back from the main road or whatever).

But given those assumptions, your probability of having a neighbor if you purchase a (randomly selected) condo can be calculated as follows. Your probability of buying an end condo is 2/2n = 1/n, and your probability of buying a middle condo is (n-1)/n. If you buy an end condo, the probability your one neighboring condo is occupied is n/(2n-1). If you buy a middle condo, the probability neither of the neighboring condos is occupied is [(n-1)*(n-2)] / [(2n-1)*(2n-2)], so the probability at least one of the neighboring condos is occupied (ie you have a neighbor) is 1 - [(n-1)*(n-2)] / [(2n-1)*(2n-2)].

So your probability of having a neighbor is:
(1/n) * (n/(2n-1)) +
((n-1)/n) * [((2n-1)*(2n-2) - (n-1)*(n-2)) / ((2n-1)*(2n-2))]

Whatever that works out to but yes, it will always be > 50%.
And I have no idea what this has to do with the Monty Hall problem.

Thanks for the analysis, but you don't see the connection? Or maybe I'm drawing a thread where there is none? I don't know enough to say for sure.

My gut said it had to be >50%. Of course math > gut when it comes to something like this.

About 3/4 when n is large. Heuristically, chance of no neighbor on the left is about 1/2, same for the right, so chance of no neighbors on either side is about 1/4.

No, I don't see it either. Why don't you explain the thread to see what's going on?

The only connection I've been able to make so far is that there is an "expected probability" based on a faulty cursory analysis (that you would expect a 50% chance of having a neighbor), but the actual probability is different.

Yeah, come to think of it, I jumped the gun. They're not even close. I'm usually better at correcting myself and not starting threads being totally wrong or totally "out there" whenever some question pops. Else, there would be a barrage of low quality threads and I'd get banned.

I was thinking along the lines of the odds from the condo owner's perspective compared to an outsider's perspective. There is no difference. Faulty line of thinking there.

I have no idea why I even confused that as having similarities with the Monty Hall problem. Maybe it had something to with the differences in the odds having incomplete, partial and complete information. Complete information would be from the perspective of being whoever has special knowledge e.g., being the person behind the door in the Monty Hall problem who gets revealed knowing that one of the other two doors has the real person (assuming this version is about people and not objects).

Compare this to the guy who knows he's the neighbour and knows that the guy to his right (or left) - middle-guy- doesn't know which side he's on. See, the neighbour KNOWS the "chance" from the middle-guy's perspective is... whatever it was (~3/4? yeah).

Ah.. That was fun. OK, thread can die peacefully now, please. It served its purpose, I think.